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I recently learned that there is a one-to-one correspondence between isomorphism classes of complex reductive groups and isomorphism classes of compact connected real Lie groups given by taking a compact real form in one direction and complexification in the other direction. I am happy with many parallels such as the existence of maximal tori, the definition of a Weyl groups, the classification of finite dimensional irreducible representations by dominant integral weights and so on. (Please correct me if I get any of the statements above wrong since I am just pulling them out from my memory.)

One question that bothers me is the following. In the complex reductive case, given a fixed maximal torus one can choose a Borel subgroup containing that torus, which is equivalent to choosing a collection of simple roots in the root system; is there an analog of this in the compact connected real Lie group case? (I don't know how to properly define a root system for a compact connected real Lie group without passing to the corresponding complex reductive group either.) Thanks a lot.

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    $\begingroup$ the analogue of a Borel subgroup is a maximal torus in the compact real form. Indeed, $G/T\simeq G(\mathbb{C})/B$ for any Borel subgroup containing $T$, where $T$ is a maximal torus of $G$, the compact real form of the complex group $G(\mathbb{C})$. $\endgroup$ – Venkataramana Mar 15 '17 at 1:52
  • $\begingroup$ @Venkataramana I see. So there is no analogue of the choices of different $B$ containing the same $T$ in the compact real form then, since any $B$ containing $T$ would make the isomorphism $G/T\cong G(\mathbb{C})/B$ hold. This is interesting. Thanks. $\endgroup$ – Daps Mar 15 '17 at 2:00
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    $\begingroup$ One doesn't need to introduce complex reductive groups, just the complexified Lie algebra equipped with its action by a maximal torus $T$ of the connected compact group $K$: in the classical theory one proves directly that the set of non-trivial $T$-weights on ${\rm{Lie}}(K)_{\mathbf{C}}$ is a root system in its $\mathbf{Q}$-span inside the rationalized character lattice ${\rm{X}}(T)_{\mathbf{Q}}$ (whose associated real vector space is canonically identified with the linear dual of ${\rm{Lie}}(T)$) and has all desired properties. See the book of Brocker and tom Dieck on compact groups. $\endgroup$ – nfdc23 Mar 15 '17 at 2:57
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    $\begingroup$ Note that the root lines inside the complexified Lie algebra don't arise as Lie algebras of subgroups of the (connected) compact group either, in contrast with the case of complex reductive groups. This is analogous to the issue with a Borel subalgebra of the complexified Lie algebra not arising from a subgroup of the compact group. In effect, working with the rich structure of the complexified Lie algebra when studying the structure of the connected compact group is like an out-of-body experience. $\endgroup$ – nfdc23 Mar 15 '17 at 3:00
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As the comments indicate, you have to go over to the complexified Lie algebra to study the simple roots and corresponding Borel subalgebras. This is a traditional pathway, so I'll add (in community-wiki format) a couple of explicit references. One was suggested already: the translated text by Brocker and tom Dieck, which covers both the structure theory and the (finite dimensional!) representation theory of a compact Lie group $G$ here.

Early in the book they work directly with $G$, showing for example that each element lies in a conjugate of a fixed maximal torus $T$. (In algebraic group language, every element of $G$ is "semisimple".) But eventually the root system has to be studied using the complexified Lie algebra, with results brought back to the compact Lie groups.

For a concise but more formal treatment in a similar spirit, you might want to compare Chapter 9 in Bourbaki's treatise Lie Groups and Lie Algebras (original French edition, 1982).

As Venkataramana observes, the flag manifold in the compact setting is a homogeneous space $G/T$, but in the complex Lie group setting it is realized as $G_\mathbb{C}/B$ for a Borel subgroup $B$. Either way, the construction of finite dimensional irreducible representations is often carried out in terms of global sections of line bundles on this manifold in the spirit of Borel-Weil, relative to weights in a dominant Weyl chamber.

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It sounds like you want a datum to associate to a compact Lie group and chosen torus, that's functorially the same as a choice of Borel containing that torus if one were to complexify, without using the word "complexify" anywhere.

If that's right, I suggest "choice of connected component of $(\mathfrak t^*$ with all points removed that have nontrivial $W$-stabilizer)".

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